A number line is numbered with the integers from 0 to 50 inclusive. An

Question

A number line is numbered with the integers from 0 to 50 inclusive. An ant walks along the number line as follows: First, it walks in the positive direction until it reaches a multiple of 5 that it hasn’t previously reached. Then, it walks in the negative direction for at least 1 unit, stopping when it reaches any multiple of 2. The ant starts at 0 on the number line and repeats this process until it first reaches the point marked 50. How many units does it travel in total?

(A) 60

(B) 65

(C) 76

(D) 80

(E) 90

(A) 60

(B) 65

(C) 76

(D) 80

(E) 90

IanStewart · Accepted Answer

Every time the ant reaches an odd multiple of five, it will travel 2 extra units -- 1 unit to go back to the even number just below the odd multiple of five, and 1 unit to go forward again to the odd multiple of five. Similarly, every time it reaches an even multiple of five, it will travel 4 extra units, since it must now go back two units to the nearest even number, then two units forward again. Since there are 5 odd multiples of five, and 4 even multiples of five, on this number line, the ant travels an additional 5*2 + 4*4 = 26 units, in addition to the 50 units it travels from 0 to 50, for a total of 76 units.

gustavvp · Answer

I did this manually (took about 1m10s), is there another faster way to do this?

karnavora1 · Answer

There is pattern to this  gustavvp

Some people will kill me for saying this but doing it manually is also okay

But otherwise once you start counting you will realize for every number after 5 he is walking steps 8 steps (excluding the last one where he stops at 50)

So if you add the first 6 steps (0 to 5 and again 5 to 4) to (8*9)-2 you will get 76.

I hope I was clear.

Blair15 · Answer

Can someone please explain this?

av1901 · Answer

Sure.
So, as per the question: Positive direction implies movement from left to right on the number line and negative direction implies movement from right to left on the number line

So the ant travels to a multiple of 5 which it has not previously reached, so we can say that  it goes to 5 in its first instance 
Then it goes in negative direction by at least 1 unit to a number which is a multiple of 2 so in this instance,  it goes to 4

So, now after this, it will go the next multiple of 5 which it has not reached,  which is 10 , and then back in negative  to the number 8 (because 9 is not a multiple of 2)

So, we can see that there is a pattern forming here in terms of the ant's movement on the number line

5--4--10--8--15--14--20--18--25--24--30--28--35--34--40--38--45--44--50

We can observe that after reaching 5, the ant travels either 7 or 9 alternately to reach the next multiple of 5

So, total units travelled = 5 + (7+9+7+9+7+9+7+9+7) = 76

Answer - C

samarpan.g28 · Answer

Below are the numbers where the ants will step on.
5,4,10,8,15,14,20,18,25,24,30,28,35,34,40,38,45,44,50.

If we calculate the steps between the numbers, they will be as below.
5,1,6,2,7,1,6,2,7,1,6,2,7,1,6,2,7,1,6 - notice the pattern: (5,1,6,2), (7,1,6,2), (7,1,6,2), (7,1,6,2), (7,1,6).

Therefore, (5+1+6+2) + 3*(7+1+6+2) + (7+1+6)=14+48+14=76. Option (C) is correct.

onlymalapink · Answer

whyb would we not count going back to 0?
I assumed going to 0 and repeating implies that we loop back.

Krunaal · Answer

Why would you go back to 0?

The question mentions that the ant goes back only until it reaches the  first multiple of 2 , i.e. once it reaches 5, it goes back to 4; once it reaches 10, it goes back to 8, and so on.

A number line is numbered with the integers from 0 to 50 inclusive. An

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A number line is numbered with the integers from 0 to 50 inclusive. An

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